Geometric Modeling of EHT Data
Comrade has been designed to work with the EHT and ngEHT. In this tutorial, we will show how to reproduce some of the results from EHTC VI 2019.
In EHTC VI, they considered fitting simple geometric models to the data to estimate the black hole's image size, shape, brightness profile, etc. In this tutorial, we will construct a similar model and fit it to the data in under 50 lines of code (sans comments). To start, we load Comrade and some other packages we need.
using ComradeLoad the Data
using Pyehtim Activating project at `~/work/Comrade.jl/Comrade.jl/examples`For reproducibility we use a stable random number genreator
using StableRNGs
rng = StableRNG(42)StableRNGs.LehmerRNG(state=0x00000000000000000000000000000055)The next step is to load the data. We will use the publically available M 87 data which can be downloaded from cyverse. For an introduction to data loading, see Loading Data into Comrade.
obs = load_uvfits_and_array(joinpath(dirname(pathof(Comrade)), "..", "examples", "SR1_M87_2017_096_lo_hops_netcal_StokesI.uvfits"))Python: <ehtim.obsdata.Obsdata object at 0x7fc6aa6e2dd0>Now we will kill 0-baselines since we don't care about large-scale flux and since we know that the gains in this dataset are coherent across a scan, we make scan-average data
obs = Pyehtim.scan_average(obs.flag_uvdist(uv_min=0.1e9))Python: <ehtim.obsdata.Obsdata object at 0x7fc6aa6e1cf0>Now we extract the data products we want to fit
dlcamp, dcphase = extract_table(obs, LogClosureAmplitudes(;snrcut=3.0), ClosurePhases(;snrcut=3.0))(EHTObservation{Float64,Comrade.EHTLogClosureAmplitudeDatum{Float64}, ...}
source: M87
mjd: 57849
frequency: 2.27070703125e11
bandwidth: 1.856e9
stations: [:AA, :AP, :AZ, :JC, :LM, :PV, :SM]
nsamples: 94
, EHTObservation{Float64,Comrade.EHTClosurePhaseDatum{Float64}, ...}
source: M87
mjd: 57849
frequency: 2.27070703125e11
bandwidth: 1.856e9
stations: [:AA, :AP, :AZ, :JC, :LM, :PV, :SM]
nsamples: 119
)!!!warn We remove the low-snr closures since they are very non-gaussian. This can create rather large biases in the model fitting since the likelihood has much heavier tails that the usual Gaussian approximation.
For the image model, we will use a modified MRing, a infinitely thin delta ring with an azimuthal structure given by a Fourier expansion. To give the MRing some width, we will convolve the ring with a Gaussian and add an additional gaussian to the image to model any non-ring flux. Comrade expects that any model function must accept a named tuple and returns must always return an object that implements the VLBISkyModels Interface
function model(θ)
(;radius, width, ma, mp, τ, ξτ, f, σG, τG, ξG, xG, yG) = θ
α = ma.*cos.(mp .- ξτ)
β = ma.*sin.(mp .- ξτ)
ring = f*smoothed(modify(MRing(α, β), Stretch(radius, radius*(1+τ)), Rotate(ξτ)), width)
g = (1-f)*shifted(rotated(stretched(Gaussian(), σG, σG*(1+τG)), ξG), xG, yG)
return ring + g
endmodel (generic function with 1 method)To construct our likelihood p(V|M) where V is our data and M is our model, we use the RadioLikelihood function. The first argument of RadioLikelihood is always a function that constructs our Comrade model from the set of parameters θ.
lklhd = RadioLikelihood(model, dlcamp, dcphase)RadioLikelihood
Number of data products: 2
We now need to specify the priors for our model. The easiest way to do this is to specify a NamedTuple of distributions:
using Distributions, VLBIImagePriors
prior = NamedDist(
radius = Uniform(μas2rad(10.0), μas2rad(30.0)),
width = Uniform(μas2rad(1.0), μas2rad(10.0)),
ma = (Uniform(0.0, 0.5), Uniform(0.0, 0.5)),
mp = (Uniform(0, 2π), Uniform(0, 2π)),
τ = Uniform(0.0, 1.0),
ξτ= Uniform(0.0, π),
f = Uniform(0.0, 1.0),
σG = Uniform(μas2rad(1.0), μas2rad(100.0)),
τG = Uniform(0.0, 1.0),
ξG = Uniform(0.0, 1π),
xG = Uniform(-μas2rad(80.0), μas2rad(80.0)),
yG = Uniform(-μas2rad(80.0), μas2rad(80.0))
)VLBIImagePriors.NamedDist{(:radius, :width, :ma, :mp, :τ, :ξτ, :f, :σG, :τG, :ξG, :xG, :yG), Tuple{Distributions.Uniform{Float64}, Distributions.Uniform{Float64}, VLBIImagePriors.TupleDist{2, Tuple{Distributions.Uniform{Float64}, Distributions.Uniform{Float64}}}, VLBIImagePriors.TupleDist{2, Tuple{Distributions.Uniform{Float64}, Distributions.Uniform{Float64}}}, Vararg{Distributions.Uniform{Float64}, 8}}}(
dists: (Distributions.Uniform{Float64}(a=4.84813681109536e-11, b=1.454441043328608e-10), Distributions.Uniform{Float64}(a=4.84813681109536e-12, b=4.84813681109536e-11), VLBIImagePriors.TupleDist{2, Tuple{Distributions.Uniform{Float64}, Distributions.Uniform{Float64}}}(dists=(Distributions.Uniform{Float64}(a=0.0, b=0.5), Distributions.Uniform{Float64}(a=0.0, b=0.5))), VLBIImagePriors.TupleDist{2, Tuple{Distributions.Uniform{Float64}, Distributions.Uniform{Float64}}}(dists=(Distributions.Uniform{Float64}(a=0.0, b=6.283185307179586), Distributions.Uniform{Float64}(a=0.0, b=6.283185307179586))), Distributions.Uniform{Float64}(a=0.0, b=1.0), Distributions.Uniform{Float64}(a=0.0, b=3.141592653589793), Distributions.Uniform{Float64}(a=0.0, b=1.0), Distributions.Uniform{Float64}(a=4.84813681109536e-12, b=4.84813681109536e-10), Distributions.Uniform{Float64}(a=0.0, b=1.0), Distributions.Uniform{Float64}(a=0.0, b=3.141592653589793), Distributions.Uniform{Float64}(a=-3.878509448876288e-10, b=3.878509448876288e-10), Distributions.Uniform{Float64}(a=-3.878509448876288e-10, b=3.878509448876288e-10))
)
Note that for α and β we use a product distribution to signify that we want to use a multivariate uniform for the mring components α and β. In general the structure of the variables is specified by the prior. Note that this structure must be compatible with the model definition model(θ).
To form the posterior we now call
post = Posterior(lklhd, prior)Posterior{RadioLikelihood{typeof(Main.model), Nothing, Tuple{Comrade.EHTObservation{Float64, Comrade.EHTLogClosureAmplitudeDatum{Float64}, StructArrays.StructVector{Comrade.EHTLogClosureAmplitudeDatum{Float64}, NamedTuple{(:measurement, :error, :U1, :V1, :U2, :V2, :U3, :V3, :U4, :V4, :T, :F, :quadrangle), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{NTuple{4, Symbol}}}}, Int64}, Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64}, Comrade.EHTObservation{Float64, Comrade.EHTClosurePhaseDatum{Float64}, StructArrays.StructVector{Comrade.EHTClosurePhaseDatum{Float64}, NamedTuple{(:measurement, :error, :U1, :V1, :U2, :V2, :U3, :V3, :T, :F, :triangle), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol, Symbol}}}}, Int64}, Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64}}, Tuple{Comrade.ConditionedLikelihood{Comrade.var"#34#35"{Float64, Base.Fix2{typeof(logclosure_amplitudes), Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}}, VLBILikelihoods.CholeskyFactor{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, SparseArrays.CHOLMOD.Factor{Float64}}}, Vector{Float64}}, Comrade.ConditionedLikelihood{Comrade.var"#36#37"{Float64, Base.Fix2{typeof(closure_phases), Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}}, VLBILikelihoods.CholeskyFactor{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, SparseArrays.CHOLMOD.Factor{Float64}}}, Vector{Float64}}}, Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}, NamedTuple{(:U, :V, :T, :F), NTuple{4, Vector{Float64}}}}, VLBIImagePriors.NamedDist{(:radius, :width, :ma, :mp, :τ, :ξτ, :f, :σG, :τG, :ξG, :xG, :yG), Tuple{Distributions.Uniform{Float64}, Distributions.Uniform{Float64}, VLBIImagePriors.TupleDist{2, Tuple{Distributions.Uniform{Float64}, Distributions.Uniform{Float64}}}, VLBIImagePriors.TupleDist{2, Tuple{Distributions.Uniform{Float64}, Distributions.Uniform{Float64}}}, Vararg{Distributions.Uniform{Float64}, 8}}}}(RadioLikelihood
Number of data products: 2
, VLBIImagePriors.NamedDist{(:radius, :width, :ma, :mp, :τ, :ξτ, :f, :σG, :τG, :ξG, :xG, :yG), Tuple{Distributions.Uniform{Float64}, Distributions.Uniform{Float64}, VLBIImagePriors.TupleDist{2, Tuple{Distributions.Uniform{Float64}, Distributions.Uniform{Float64}}}, VLBIImagePriors.TupleDist{2, Tuple{Distributions.Uniform{Float64}, Distributions.Uniform{Float64}}}, Vararg{Distributions.Uniform{Float64}, 8}}}(
dists: (Distributions.Uniform{Float64}(a=4.84813681109536e-11, b=1.454441043328608e-10), Distributions.Uniform{Float64}(a=4.84813681109536e-12, b=4.84813681109536e-11), VLBIImagePriors.TupleDist{2, Tuple{Distributions.Uniform{Float64}, Distributions.Uniform{Float64}}}(dists=(Distributions.Uniform{Float64}(a=0.0, b=0.5), Distributions.Uniform{Float64}(a=0.0, b=0.5))), VLBIImagePriors.TupleDist{2, Tuple{Distributions.Uniform{Float64}, Distributions.Uniform{Float64}}}(dists=(Distributions.Uniform{Float64}(a=0.0, b=6.283185307179586), Distributions.Uniform{Float64}(a=0.0, b=6.283185307179586))), Distributions.Uniform{Float64}(a=0.0, b=1.0), Distributions.Uniform{Float64}(a=0.0, b=3.141592653589793), Distributions.Uniform{Float64}(a=0.0, b=1.0), Distributions.Uniform{Float64}(a=4.84813681109536e-12, b=4.84813681109536e-10), Distributions.Uniform{Float64}(a=0.0, b=1.0), Distributions.Uniform{Float64}(a=0.0, b=3.141592653589793), Distributions.Uniform{Float64}(a=-3.878509448876288e-10, b=3.878509448876288e-10), Distributions.Uniform{Float64}(a=-3.878509448876288e-10, b=3.878509448876288e-10))
)
)!!!warn As of Comrade 0.9 we have switched to the proper covariant closure likelihood. This is slower than the naieve diagonal liklelihood, but takes into account the correlations between closures that share the same baselines.
This constructs a posterior density that can be evaluated by calling logdensityof. For example,
logdensityof(post, (radius = μas2rad(20.0),
width = μas2rad(10.0),
ma = (0.3, 0.3),
mp = (π/2, π),
τ = 0.1,
ξτ= π/2,
f = 0.6,
σG = μas2rad(50.0),
τG = 0.1,
ξG = 0.5,
xG = 0.0,
yG = 0.0))-7439.968456427528Reconstruction
Now that we have fully specified our model, we now will try to find the optimal reconstruction of our model given our observed data.
Currently, post is in parameter space. Often optimization and sampling algorithms want it in some modified space. For example, nested sampling algorithms want the parameters in the unit hypercube. To transform the posterior to the unit hypercube, we can use the ascube function
cpost = ascube(post)Comrade.TransformedPosterior{Posterior{RadioLikelihood{typeof(Main.model), Nothing, Tuple{Comrade.EHTObservation{Float64, Comrade.EHTLogClosureAmplitudeDatum{Float64}, StructArrays.StructVector{Comrade.EHTLogClosureAmplitudeDatum{Float64}, NamedTuple{(:measurement, :error, :U1, :V1, :U2, :V2, :U3, :V3, :U4, :V4, :T, :F, :quadrangle), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{NTuple{4, Symbol}}}}, Int64}, Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64}, Comrade.EHTObservation{Float64, Comrade.EHTClosurePhaseDatum{Float64}, StructArrays.StructVector{Comrade.EHTClosurePhaseDatum{Float64}, NamedTuple{(:measurement, :error, :U1, :V1, :U2, :V2, :U3, :V3, :T, :F, :triangle), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol, Symbol}}}}, Int64}, Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64}}, Tuple{Comrade.ConditionedLikelihood{Comrade.var"#34#35"{Float64, Base.Fix2{typeof(logclosure_amplitudes), Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}}, VLBILikelihoods.CholeskyFactor{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, SparseArrays.CHOLMOD.Factor{Float64}}}, Vector{Float64}}, Comrade.ConditionedLikelihood{Comrade.var"#36#37"{Float64, Base.Fix2{typeof(closure_phases), Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}}, VLBILikelihoods.CholeskyFactor{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, SparseArrays.CHOLMOD.Factor{Float64}}}, Vector{Float64}}}, Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}, NamedTuple{(:U, :V, :T, :F), NTuple{4, Vector{Float64}}}}, VLBIImagePriors.NamedDist{(:radius, :width, :ma, :mp, :τ, :ξτ, :f, :σG, :τG, :ξG, :xG, :yG), Tuple{Distributions.Uniform{Float64}, Distributions.Uniform{Float64}, VLBIImagePriors.TupleDist{2, Tuple{Distributions.Uniform{Float64}, Distributions.Uniform{Float64}}}, VLBIImagePriors.TupleDist{2, Tuple{Distributions.Uniform{Float64}, Distributions.Uniform{Float64}}}, Vararg{Distributions.Uniform{Float64}, 8}}}}, HypercubeTransform.TupleHC{NamedTuple{(:radius, :width, :ma, :mp, :τ, :ξτ, :f, :σG, :τG, :ξG, :xG, :yG), Tuple{HypercubeTransform.ScalarHC{Distributions.Uniform{Float64}}, HypercubeTransform.ScalarHC{Distributions.Uniform{Float64}}, HypercubeTransform.TupleHC{Tuple{HypercubeTransform.ScalarHC{Distributions.Uniform{Float64}}, HypercubeTransform.ScalarHC{Distributions.Uniform{Float64}}}}, HypercubeTransform.TupleHC{Tuple{HypercubeTransform.ScalarHC{Distributions.Uniform{Float64}}, HypercubeTransform.ScalarHC{Distributions.Uniform{Float64}}}}, Vararg{HypercubeTransform.ScalarHC{Distributions.Uniform{Float64}}, 8}}}}}(Posterior{RadioLikelihood{typeof(Main.model), Nothing, Tuple{Comrade.EHTObservation{Float64, Comrade.EHTLogClosureAmplitudeDatum{Float64}, StructArrays.StructVector{Comrade.EHTLogClosureAmplitudeDatum{Float64}, NamedTuple{(:measurement, :error, :U1, :V1, :U2, :V2, :U3, :V3, :U4, :V4, :T, :F, :quadrangle), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{NTuple{4, Symbol}}}}, Int64}, Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64}, Comrade.EHTObservation{Float64, Comrade.EHTClosurePhaseDatum{Float64}, StructArrays.StructVector{Comrade.EHTClosurePhaseDatum{Float64}, NamedTuple{(:measurement, :error, :U1, :V1, :U2, :V2, :U3, :V3, :T, :F, :triangle), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol, Symbol}}}}, Int64}, Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64}}, Tuple{Comrade.ConditionedLikelihood{Comrade.var"#34#35"{Float64, Base.Fix2{typeof(logclosure_amplitudes), Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}}, VLBILikelihoods.CholeskyFactor{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, SparseArrays.CHOLMOD.Factor{Float64}}}, Vector{Float64}}, Comrade.ConditionedLikelihood{Comrade.var"#36#37"{Float64, Base.Fix2{typeof(closure_phases), Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}}, VLBILikelihoods.CholeskyFactor{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, SparseArrays.CHOLMOD.Factor{Float64}}}, Vector{Float64}}}, Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}, NamedTuple{(:U, :V, :T, :F), NTuple{4, Vector{Float64}}}}, VLBIImagePriors.NamedDist{(:radius, :width, :ma, :mp, :τ, :ξτ, :f, :σG, :τG, :ξG, :xG, :yG), Tuple{Distributions.Uniform{Float64}, Distributions.Uniform{Float64}, VLBIImagePriors.TupleDist{2, Tuple{Distributions.Uniform{Float64}, Distributions.Uniform{Float64}}}, VLBIImagePriors.TupleDist{2, Tuple{Distributions.Uniform{Float64}, Distributions.Uniform{Float64}}}, Vararg{Distributions.Uniform{Float64}, 8}}}}(RadioLikelihood
Number of data products: 2
, VLBIImagePriors.NamedDist{(:radius, :width, :ma, :mp, :τ, :ξτ, :f, :σG, :τG, :ξG, :xG, :yG), Tuple{Distributions.Uniform{Float64}, Distributions.Uniform{Float64}, VLBIImagePriors.TupleDist{2, Tuple{Distributions.Uniform{Float64}, Distributions.Uniform{Float64}}}, VLBIImagePriors.TupleDist{2, Tuple{Distributions.Uniform{Float64}, Distributions.Uniform{Float64}}}, Vararg{Distributions.Uniform{Float64}, 8}}}(
dists: (Distributions.Uniform{Float64}(a=4.84813681109536e-11, b=1.454441043328608e-10), Distributions.Uniform{Float64}(a=4.84813681109536e-12, b=4.84813681109536e-11), VLBIImagePriors.TupleDist{2, Tuple{Distributions.Uniform{Float64}, Distributions.Uniform{Float64}}}(dists=(Distributions.Uniform{Float64}(a=0.0, b=0.5), Distributions.Uniform{Float64}(a=0.0, b=0.5))), VLBIImagePriors.TupleDist{2, Tuple{Distributions.Uniform{Float64}, Distributions.Uniform{Float64}}}(dists=(Distributions.Uniform{Float64}(a=0.0, b=6.283185307179586), Distributions.Uniform{Float64}(a=0.0, b=6.283185307179586))), Distributions.Uniform{Float64}(a=0.0, b=1.0), Distributions.Uniform{Float64}(a=0.0, b=3.141592653589793), Distributions.Uniform{Float64}(a=0.0, b=1.0), Distributions.Uniform{Float64}(a=4.84813681109536e-12, b=4.84813681109536e-10), Distributions.Uniform{Float64}(a=0.0, b=1.0), Distributions.Uniform{Float64}(a=0.0, b=3.141592653589793), Distributions.Uniform{Float64}(a=-3.878509448876288e-10, b=3.878509448876288e-10), Distributions.Uniform{Float64}(a=-3.878509448876288e-10, b=3.878509448876288e-10))
)
), HypercubeTransform.TupleHC{NamedTuple{(:radius, :width, :ma, :mp, :τ, :ξτ, :f, :σG, :τG, :ξG, :xG, :yG), Tuple{HypercubeTransform.ScalarHC{Distributions.Uniform{Float64}}, HypercubeTransform.ScalarHC{Distributions.Uniform{Float64}}, HypercubeTransform.TupleHC{Tuple{HypercubeTransform.ScalarHC{Distributions.Uniform{Float64}}, HypercubeTransform.ScalarHC{Distributions.Uniform{Float64}}}}, HypercubeTransform.TupleHC{Tuple{HypercubeTransform.ScalarHC{Distributions.Uniform{Float64}}, HypercubeTransform.ScalarHC{Distributions.Uniform{Float64}}}}, Vararg{HypercubeTransform.ScalarHC{Distributions.Uniform{Float64}}, 8}}}}((radius = HypercubeTransform.ScalarHC{Distributions.Uniform{Float64}}(Distributions.Uniform{Float64}(a=4.84813681109536e-11, b=1.454441043328608e-10)), width = HypercubeTransform.ScalarHC{Distributions.Uniform{Float64}}(Distributions.Uniform{Float64}(a=4.84813681109536e-12, b=4.84813681109536e-11)), ma = HypercubeTransform.TupleHC{Tuple{HypercubeTransform.ScalarHC{Distributions.Uniform{Float64}}, HypercubeTransform.ScalarHC{Distributions.Uniform{Float64}}}}((HypercubeTransform.ScalarHC{Distributions.Uniform{Float64}}(Distributions.Uniform{Float64}(a=0.0, b=0.5)), HypercubeTransform.ScalarHC{Distributions.Uniform{Float64}}(Distributions.Uniform{Float64}(a=0.0, b=0.5))), 2), mp = HypercubeTransform.TupleHC{Tuple{HypercubeTransform.ScalarHC{Distributions.Uniform{Float64}}, HypercubeTransform.ScalarHC{Distributions.Uniform{Float64}}}}((HypercubeTransform.ScalarHC{Distributions.Uniform{Float64}}(Distributions.Uniform{Float64}(a=0.0, b=6.283185307179586)), HypercubeTransform.ScalarHC{Distributions.Uniform{Float64}}(Distributions.Uniform{Float64}(a=0.0, b=6.283185307179586))), 2), τ = HypercubeTransform.ScalarHC{Distributions.Uniform{Float64}}(Distributions.Uniform{Float64}(a=0.0, b=1.0)), ξτ = HypercubeTransform.ScalarHC{Distributions.Uniform{Float64}}(Distributions.Uniform{Float64}(a=0.0, b=3.141592653589793)), f = HypercubeTransform.ScalarHC{Distributions.Uniform{Float64}}(Distributions.Uniform{Float64}(a=0.0, b=1.0)), σG = HypercubeTransform.ScalarHC{Distributions.Uniform{Float64}}(Distributions.Uniform{Float64}(a=4.84813681109536e-12, b=4.84813681109536e-10)), τG = HypercubeTransform.ScalarHC{Distributions.Uniform{Float64}}(Distributions.Uniform{Float64}(a=0.0, b=1.0)), ξG = HypercubeTransform.ScalarHC{Distributions.Uniform{Float64}}(Distributions.Uniform{Float64}(a=0.0, b=3.141592653589793)), xG = HypercubeTransform.ScalarHC{Distributions.Uniform{Float64}}(Distributions.Uniform{Float64}(a=-3.878509448876288e-10, b=3.878509448876288e-10)), yG = HypercubeTransform.ScalarHC{Distributions.Uniform{Float64}}(Distributions.Uniform{Float64}(a=-3.878509448876288e-10, b=3.878509448876288e-10))), 14))If we want to flatten the parameter space and move from constrained parameters to (-∞, ∞) support we can use the asflat function
fpost = asflat(post)Comrade.TransformedPosterior{Posterior{RadioLikelihood{typeof(Main.model), Nothing, Tuple{Comrade.EHTObservation{Float64, Comrade.EHTLogClosureAmplitudeDatum{Float64}, StructArrays.StructVector{Comrade.EHTLogClosureAmplitudeDatum{Float64}, NamedTuple{(:measurement, :error, :U1, :V1, :U2, :V2, :U3, :V3, :U4, :V4, :T, :F, :quadrangle), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{NTuple{4, Symbol}}}}, Int64}, Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64}, Comrade.EHTObservation{Float64, Comrade.EHTClosurePhaseDatum{Float64}, StructArrays.StructVector{Comrade.EHTClosurePhaseDatum{Float64}, NamedTuple{(:measurement, :error, :U1, :V1, :U2, :V2, :U3, :V3, :T, :F, :triangle), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol, Symbol}}}}, Int64}, Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64}}, Tuple{Comrade.ConditionedLikelihood{Comrade.var"#34#35"{Float64, Base.Fix2{typeof(logclosure_amplitudes), Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}}, VLBILikelihoods.CholeskyFactor{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, SparseArrays.CHOLMOD.Factor{Float64}}}, Vector{Float64}}, Comrade.ConditionedLikelihood{Comrade.var"#36#37"{Float64, Base.Fix2{typeof(closure_phases), Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}}, VLBILikelihoods.CholeskyFactor{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, SparseArrays.CHOLMOD.Factor{Float64}}}, Vector{Float64}}}, Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}, NamedTuple{(:U, :V, :T, :F), NTuple{4, Vector{Float64}}}}, VLBIImagePriors.NamedDist{(:radius, :width, :ma, :mp, :τ, :ξτ, :f, :σG, :τG, :ξG, :xG, :yG), Tuple{Distributions.Uniform{Float64}, Distributions.Uniform{Float64}, VLBIImagePriors.TupleDist{2, Tuple{Distributions.Uniform{Float64}, Distributions.Uniform{Float64}}}, VLBIImagePriors.TupleDist{2, Tuple{Distributions.Uniform{Float64}, Distributions.Uniform{Float64}}}, Vararg{Distributions.Uniform{Float64}, 8}}}}, TransformVariables.TransformTuple{NamedTuple{(:radius, :width, :ma, :mp, :τ, :ξτ, :f, :σG, :τG, :ξG, :xG, :yG), Tuple{TransformVariables.ScaledShiftedLogistic{Float64}, TransformVariables.ScaledShiftedLogistic{Float64}, TransformVariables.TransformTuple{Tuple{TransformVariables.ScaledShiftedLogistic{Float64}, TransformVariables.ScaledShiftedLogistic{Float64}}}, TransformVariables.TransformTuple{Tuple{TransformVariables.ScaledShiftedLogistic{Float64}, TransformVariables.ScaledShiftedLogistic{Float64}}}, Vararg{TransformVariables.ScaledShiftedLogistic{Float64}, 8}}}}}(Posterior{RadioLikelihood{typeof(Main.model), Nothing, Tuple{Comrade.EHTObservation{Float64, Comrade.EHTLogClosureAmplitudeDatum{Float64}, StructArrays.StructVector{Comrade.EHTLogClosureAmplitudeDatum{Float64}, NamedTuple{(:measurement, :error, :U1, :V1, :U2, :V2, :U3, :V3, :U4, :V4, :T, :F, :quadrangle), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{NTuple{4, Symbol}}}}, Int64}, Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64}, Comrade.EHTObservation{Float64, Comrade.EHTClosurePhaseDatum{Float64}, StructArrays.StructVector{Comrade.EHTClosurePhaseDatum{Float64}, NamedTuple{(:measurement, :error, :U1, :V1, :U2, :V2, :U3, :V3, :T, :F, :triangle), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol, Symbol}}}}, Int64}, Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64}}, Tuple{Comrade.ConditionedLikelihood{Comrade.var"#34#35"{Float64, Base.Fix2{typeof(logclosure_amplitudes), Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}}, VLBILikelihoods.CholeskyFactor{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, SparseArrays.CHOLMOD.Factor{Float64}}}, Vector{Float64}}, Comrade.ConditionedLikelihood{Comrade.var"#36#37"{Float64, Base.Fix2{typeof(closure_phases), Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}}, VLBILikelihoods.CholeskyFactor{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, SparseArrays.CHOLMOD.Factor{Float64}}}, Vector{Float64}}}, Comrade.ClosureConfig{Comrade.EHTObservation{Float64, Comrade.EHTVisibilityDatum{Float64}, StructArrays.StructVector{Comrade.EHTVisibilityDatum{Float64}, NamedTuple{(:measurement, :error, :U, :V, :T, :F, :baseline), Tuple{Vector{ComplexF64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}}}, Int64}, Comrade.EHTArrayConfiguration{Float64, TypedTables.Table{NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Symbol, Vararg{Float64, 8}}}, 1, NamedTuple{(:sites, :X, :Y, :Z, :SEFD1, :SEFD2, :fr_parallactic, :fr_elevation, :fr_offset), Tuple{Vector{Symbol}, Vararg{Vector{Float64}, 8}}}}, TypedTables.Table{NamedTuple{(:start, :stop), Tuple{Float64, Float64}}, 1, NamedTuple{(:start, :stop), Tuple{Vector{Float64}, Vector{Float64}}}}, StructArrays.StructVector{Comrade.ArrayBaselineDatum, NamedTuple{(:U, :V, :T, :F, :baseline, :error, :elevation, :parallactic), Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Tuple{Symbol, Symbol}}, Vector{Float64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}, StructArrays.StructVector{Tuple{Float64, Float64}, Tuple{Vector{Float64}, Vector{Float64}}, Int64}}}, Int64}}, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}, NamedTuple{(:U, :V, :T, :F), NTuple{4, Vector{Float64}}}}, VLBIImagePriors.NamedDist{(:radius, :width, :ma, :mp, :τ, :ξτ, :f, :σG, :τG, :ξG, :xG, :yG), Tuple{Distributions.Uniform{Float64}, Distributions.Uniform{Float64}, VLBIImagePriors.TupleDist{2, Tuple{Distributions.Uniform{Float64}, Distributions.Uniform{Float64}}}, VLBIImagePriors.TupleDist{2, Tuple{Distributions.Uniform{Float64}, Distributions.Uniform{Float64}}}, Vararg{Distributions.Uniform{Float64}, 8}}}}(RadioLikelihood
Number of data products: 2
, VLBIImagePriors.NamedDist{(:radius, :width, :ma, :mp, :τ, :ξτ, :f, :σG, :τG, :ξG, :xG, :yG), Tuple{Distributions.Uniform{Float64}, Distributions.Uniform{Float64}, VLBIImagePriors.TupleDist{2, Tuple{Distributions.Uniform{Float64}, Distributions.Uniform{Float64}}}, VLBIImagePriors.TupleDist{2, Tuple{Distributions.Uniform{Float64}, Distributions.Uniform{Float64}}}, Vararg{Distributions.Uniform{Float64}, 8}}}(
dists: (Distributions.Uniform{Float64}(a=4.84813681109536e-11, b=1.454441043328608e-10), Distributions.Uniform{Float64}(a=4.84813681109536e-12, b=4.84813681109536e-11), VLBIImagePriors.TupleDist{2, Tuple{Distributions.Uniform{Float64}, Distributions.Uniform{Float64}}}(dists=(Distributions.Uniform{Float64}(a=0.0, b=0.5), Distributions.Uniform{Float64}(a=0.0, b=0.5))), VLBIImagePriors.TupleDist{2, Tuple{Distributions.Uniform{Float64}, Distributions.Uniform{Float64}}}(dists=(Distributions.Uniform{Float64}(a=0.0, b=6.283185307179586), Distributions.Uniform{Float64}(a=0.0, b=6.283185307179586))), Distributions.Uniform{Float64}(a=0.0, b=1.0), Distributions.Uniform{Float64}(a=0.0, b=3.141592653589793), Distributions.Uniform{Float64}(a=0.0, b=1.0), Distributions.Uniform{Float64}(a=4.84813681109536e-12, b=4.84813681109536e-10), Distributions.Uniform{Float64}(a=0.0, b=1.0), Distributions.Uniform{Float64}(a=0.0, b=3.141592653589793), Distributions.Uniform{Float64}(a=-3.878509448876288e-10, b=3.878509448876288e-10), Distributions.Uniform{Float64}(a=-3.878509448876288e-10, b=3.878509448876288e-10))
)
), TransformVariables.TransformTuple{NamedTuple{(:radius, :width, :ma, :mp, :τ, :ξτ, :f, :σG, :τG, :ξG, :xG, :yG), Tuple{TransformVariables.ScaledShiftedLogistic{Float64}, TransformVariables.ScaledShiftedLogistic{Float64}, TransformVariables.TransformTuple{Tuple{TransformVariables.ScaledShiftedLogistic{Float64}, TransformVariables.ScaledShiftedLogistic{Float64}}}, TransformVariables.TransformTuple{Tuple{TransformVariables.ScaledShiftedLogistic{Float64}, TransformVariables.ScaledShiftedLogistic{Float64}}}, Vararg{TransformVariables.ScaledShiftedLogistic{Float64}, 8}}}}((radius = as(Real, 4.84813681109536e-11, 1.454441043328608e-10), width = as(Real, 4.84813681109536e-12, 4.84813681109536e-11), ma = TransformVariables.TransformTuple{Tuple{TransformVariables.ScaledShiftedLogistic{Float64}, TransformVariables.ScaledShiftedLogistic{Float64}}}((as(Real, 0.0, 0.5), as(Real, 0.0, 0.5)), 2), mp = TransformVariables.TransformTuple{Tuple{TransformVariables.ScaledShiftedLogistic{Float64}, TransformVariables.ScaledShiftedLogistic{Float64}}}((as(Real, 0.0, 6.283185307179586), as(Real, 0.0, 6.283185307179586)), 2), τ = as(Real, 0.0, 1.0), ξτ = as(Real, 0.0, 3.141592653589793), f = as(Real, 0.0, 1.0), σG = as(Real, 4.84813681109536e-12, 4.84813681109536e-10), τG = as(Real, 0.0, 1.0), ξG = as(Real, 0.0, 3.141592653589793), xG = as(Real, -3.878509448876288e-10, 3.878509448876288e-10), yG = as(Real, -3.878509448876288e-10, 3.878509448876288e-10)), 14))These transformed posterior expect a vector of parameters. That is we can evaluate the transformed log density by calling
logdensityof(cpost, rand(rng, dimension(cpost)))
logdensityof(fpost, randn(rng, dimension(fpost)))-25329.634311799513note that cpost logdensity vector expects that each element lives in [0,1].
Finding the Optimal Image
Typically, most VLBI modeling codes only care about finding the optimal or best guess image of our posterior post To do this, we will use Optimization.jl and specifically the BlackBoxOptim.jl package. For Comrade, this workflow is very similar to the usual Optimization.jl workflow. The only thing to keep in mind is that Optimization.jl expects that the function we are evaluating expects the parameters to be represented as a flat Vector of float. Therefore, we must use one of our transformed posteriors, cpost or fpost. For this example
#, we will use `cpost` since it restricts the domain to live within the compact unit hypercube
#, which is easier to explore for non-gradient-based optimizers like `BBO`.
using ComradeOptimization
using OptimizationBBO
ndim = dimension(fpost)
f = OptimizationFunction(fpost)
prob = Optimization.OptimizationProblem(f, randn(rng, ndim), nothing, lb=fill(-5.0, ndim), ub=fill(5.0, ndim))OptimizationProblem. In-place: true
u0: 14-element Vector{Float64}:
-2.042002086038424
-0.7870208631180695
0.5905224676986801
0.6421092391624518
0.45523702486217854
0.05420311090968867
0.2884763107619252
0.5868282026416113
-1.6942586185480404
-0.6959809938112977
-0.30096086778202435
2.1005530576443556
-0.6896446526429071
-0.7299727780343646Now we solve for our optimial image.
sol = solve(prob, BBO_adaptive_de_rand_1_bin_radiuslimited(); maxiters=50_000);The sol vector is in the transformed space, so first we need to transform back to parameter space to that we can interpret the solution.
xopt = transform(fpost, sol)(radius = 9.844953700940736e-11, width = 2.0353993963421008e-11, ma = (0.2852037724395272, 0.09795913423243231), mp = (2.4777807280473056, 3.4105349515246735), τ = 0.0931800492127429, ξτ = 0.702012315475934, f = 0.10673907704960503, σG = 2.6776447412748347e-10, τG = 0.8716835221161645, ξG = 1.0563729402096693, xG = -2.5152780381577344e-10, yG = -1.1911798734414325e-10)Given this we can now plot the optimal image or the maximum a posteriori (MAP) image.
import WGLMakie as CM
g = imagepixels(μas2rad(200.0), μas2rad(200.0), 256, 256)
fig, ax, plt = CM.image(g, model(xopt); axis=(xreversed=true, aspect=1, xlabel="RA (μas)", ylabel="Dec (μas)"), figure=(;resolution=(650,500),) ,colormap=:afmhot)